KILLED
proof of input_3YXcv52Bt6.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 243 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 132 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 634 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (38) CdtProblem
    (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 133 ms]
    (42) CdtProblem
    (43) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 20 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms]
    (64) CdtProblem
    (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs))
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs))
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs))
   power(x, Nil) -> Cons(Nil, Nil)
   mult(x, Nil) -> Nil
   add0(x, Nil) -> x
   goal(x, y) -> power(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs))
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs))
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs))
   power(x, Nil) -> Cons(Nil, Nil)
   mult(x, Nil) -> Nil
   add0(x, Nil) -> x
   goal(x, y) -> power(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs))
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs))
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs))
   power(x, Nil) -> Cons(Nil, Nil)
   mult(x, Nil) -> Nil
   add0(x, Nil) -> x
   goal(x, y) -> power(x, y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs)) [1]
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) [1]
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1]
   power(x, Nil) -> Cons(Nil, Nil) [1]
   mult(x, Nil) -> Nil [1]
   add0(x, Nil) -> x [1]
   goal(x, y) -> power(x, y) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs)) [1]
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) [1]
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1]
   power(x, Nil) -> Cons(Nil, Nil) [1]
   mult(x, Nil) -> Nil [1]
   add0(x, Nil) -> x [1]
   goal(x, y) -> power(x, y) [1]

The TRS has the following type information:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   goal_2

(c) The following functions are completely defined:

   power_2
   mult_2
   add0_2

Due to the following rules being added:
none

And the following fresh constants: none

----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs)) [1]
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) [1]
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1]
   power(x, Nil) -> Cons(Nil, Nil) [1]
   mult(x, Nil) -> Nil [1]
   add0(x, Nil) -> x [1]
   goal(x, y) -> power(x, y) [1]

The TRS has the following type information:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(11) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   power(x', Cons(x, Cons(x1, xs'))) -> mult(x', mult(x', power(x', xs'))) [2]
   power(x', Cons(x, Nil)) -> mult(x', Cons(Nil, Nil)) [2]
   mult(x', Cons(x, Cons(x2, xs''))) -> add0(x', add0(x', mult(x', xs''))) [2]
   mult(x', Cons(x, Nil)) -> add0(x', Nil) [2]
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1]
   power(x, Nil) -> Cons(Nil, Nil) [1]
   mult(x, Nil) -> Nil [1]
   add0(x, Nil) -> x [1]
   goal(x, y) -> power(x, y) [1]

The TRS has the following type information:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(x', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   goal(z, z') -{ 1 }-> power(x, y) :|: x >= 0, y >= 0, z = x, z' = y
   mult(z, z') -{ 2 }-> add0(x', add0(x', mult(x', xs''))) :|: x' >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0, z = x'
   mult(z, z') -{ 2 }-> add0(x', 0) :|: x' >= 0, x >= 0, z' = 1 + x + 0, z = x'
   mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   power(z, z') -{ 2 }-> mult(x', mult(x', power(x', xs'))) :|: x1 >= 0, x' >= 0, x >= 0, xs' >= 0, z = x', z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(x', 1 + 0 + 0) :|: x' >= 0, x >= 0, z' = 1 + x + 0, z = x'
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: x >= 0, z = x, z' = 0


----------------------------------------

(15) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0


----------------------------------------

(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { add0 }
   { mult }
   { power }
   { goal }

----------------------------------------

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {add0}, {mult}, {power}, {goal}

----------------------------------------

(19) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {add0}, {mult}, {power}, {goal}

----------------------------------------

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + 2*z'

----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {add0}, {mult}, {power}, {goal}
Previous analysis results are:
  add0: runtime: ?, size: O(n^1) [z + 2*z']

----------------------------------------

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(z, xs) :|: xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' - 1 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {mult}, {power}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 2*z']

----------------------------------------

(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 + xs }-> 1 + (1 + 0 + 0) + s' :|: s' >= 0, s' <= z + 2 * xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0
   mult(z, z') -{ 3 }-> s :|: s >= 0, s <= z + 2 * 0, z >= 0, z' - 1 >= 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {mult}, {power}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 2*z']

----------------------------------------

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: mult
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 + xs }-> 1 + (1 + 0 + 0) + s' :|: s' >= 0, s' <= z + 2 * xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0
   mult(z, z') -{ 3 }-> s :|: s >= 0, s <= z + 2 * 0, z >= 0, z' - 1 >= 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {mult}, {power}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 2*z']
  mult: runtime: ?, size: EXP

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: mult
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 + xs }-> 1 + (1 + 0 + 0) + s' :|: s' >= 0, s' <= z + 2 * xs, xs >= 0, z' = 1 + x + xs, z >= 0, x >= 0
   goal(z, z') -{ 1 }-> power(z, z') :|: z >= 0, z' >= 0
   mult(z, z') -{ 3 }-> s :|: s >= 0, s <= z + 2 * 0, z >= 0, z' - 1 >= 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, xs''))) :|: z >= 0, x >= 0, xs'' >= 0, z' = 1 + x + (1 + x2 + xs''), x2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, xs'))) :|: x1 >= 0, z >= 0, x >= 0, xs' >= 0, z' = 1 + x + (1 + x1 + xs')
   power(z, z') -{ 2 }-> mult(z, 1 + 0 + 0) :|: z >= 0, z' - 1 >= 0
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {mult}, {power}, {goal}
Previous analysis results are:
  add0: runtime: O(n^1) [1 + z'], size: O(n^1) [z + 2*z']
  mult: runtime: INF, size: EXP

----------------------------------------

(31) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

----------------------------------------

(32)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   power(x', Cons(x, xs)) -> mult(x', power(x', xs)) [1]
   mult(x', Cons(x, xs)) -> add0(x', mult(x', xs)) [1]
   add0(x', Cons(x, xs)) -> Cons(Cons(Nil, Nil), add0(x', xs)) [1]
   power(x, Nil) -> Cons(Nil, Nil) [1]
   mult(x, Nil) -> Nil [1]
   add0(x, Nil) -> x [1]
   goal(x, y) -> power(x, y) [1]

The TRS has the following type information:
power :: Cons:Nil -> Cons:Nil -> Cons:Nil
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
mult :: Cons:Nil -> Cons:Nil -> Cons:Nil
add0 :: Cons:Nil -> Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
goal :: Cons:Nil -> Cons:Nil -> Cons:Nil

Rewrite Strategy: INNERMOST
----------------------------------------

(33) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0

----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   add0(z, z') -{ 1 }-> 1 + (1 + 0 + 0) + add0(x', xs) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   goal(z, z') -{ 1 }-> power(x, y) :|: x >= 0, y >= 0, z = x, z' = y
   mult(z, z') -{ 1 }-> add0(x', mult(x', xs)) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   power(z, z') -{ 1 }-> mult(x', power(x', xs)) :|: xs >= 0, z' = 1 + x + xs, x' >= 0, x >= 0, z = x'
   power(z, z') -{ 1 }-> 1 + 0 + 0 :|: x >= 0, z = x, z' = 0

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(35) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
   goal(z0, z1) -> power(z0, z1)
Tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   POWER(z0, Nil) -> c1
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   MULT(z0, Nil) -> c3
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   ADD0(z0, Nil) -> c5
   GOAL(z0, z1) -> c6(POWER(z0, z1))
S tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   POWER(z0, Nil) -> c1
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   MULT(z0, Nil) -> c3
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   ADD0(z0, Nil) -> c5
   GOAL(z0, z1) -> c6(POWER(z0, z1))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, goal_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2, GOAL_2

Compound Symbols:   c_2, c1, c2_2, c3, c4_1, c5, c6_1


----------------------------------------

(37) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0, z1) -> c6(POWER(z0, z1))
Removed 3 trailing nodes:
   POWER(z0, Nil) -> c1
   MULT(z0, Nil) -> c3
   ADD0(z0, Nil) -> c5

----------------------------------------

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
   goal(z0, z1) -> power(z0, z1)
Tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
S tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, goal_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c_2, c2_2, c4_1


----------------------------------------

(39) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   goal(z0, z1) -> power(z0, z1)

----------------------------------------

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
S tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c_2, c2_2, c4_1


----------------------------------------

(41) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [2] + x_1 + x_2
   POL(MULT(x_1, x_2)) = [2]
   POL(Nil) = [2]
   POL(POWER(x_1, x_2)) = [3]x_2
   POL(add0(x_1, x_2)) = [3] + [3]x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(mult(x_1, x_2)) = [2] + x_2
   POL(power(x_1, x_2)) = [2] + [3]x_1 + [3]x_2

----------------------------------------

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
S tuples:
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
K tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c_2, c2_2, c4_1


----------------------------------------

(43) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2)) by 
   POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)), POWER(z0, Nil))

----------------------------------------

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)), POWER(z0, Nil))
S tuples:
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
K tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   MULT_2, ADD0_2, POWER_2

Compound Symbols:   c2_2, c4_1, c_2


----------------------------------------

(45) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)))
S tuples:
   MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2))
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
K tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   MULT_2, ADD0_2, POWER_2

Compound Symbols:   c2_2, c4_1, c_2, c_1


----------------------------------------

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MULT(z0, Cons(z1, z2)) -> c2(ADD0(z0, mult(z0, z2)), MULT(z0, z2)) by 
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   MULT(z0, Cons(x1, Nil)) -> c2(ADD0(z0, Nil), MULT(z0, Nil))

----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   MULT(z0, Cons(x1, Nil)) -> c2(ADD0(z0, Nil), MULT(z0, Nil))
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   MULT(z0, Cons(x1, Nil)) -> c2(ADD0(z0, Nil), MULT(z0, Nil))
K tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c4_1, c_2, c_1, c2_2


----------------------------------------

(49) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   MULT(z0, Cons(x1, Nil)) -> c2
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   MULT(z0, Cons(x1, Nil)) -> c2
K tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c4_1, c_2, c_1, c2_2, c2


----------------------------------------

(51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MULT(z0, Cons(x1, Nil)) -> c2
We considered the (Usable) Rules:none
And the Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   MULT(z0, Cons(x1, Nil)) -> c2
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(MULT(x_1, x_2)) = [1]
   POL(Nil) = [1]
   POL(POWER(x_1, x_2)) = [1] + x_2
   POL(add0(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2) = 0
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(mult(x_1, x_2)) = [1] + x_2
   POL(power(x_1, x_2)) = [1] + x_1 + x_2

----------------------------------------

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   MULT(z0, Cons(x1, Nil)) -> c2
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
K tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   MULT(z0, Cons(x1, Nil)) -> c2
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c4_1, c_2, c_1, c2_2, c2


----------------------------------------

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace POWER(z0, Cons(x1, Cons(z1, z2))) -> c(MULT(z0, mult(z0, power(z0, z2))), POWER(z0, Cons(z1, z2))) by 
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))), POWER(z0, Cons(x2, Nil)))

----------------------------------------

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   MULT(z0, Cons(x1, Nil)) -> c2
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))), POWER(z0, Cons(x2, Nil)))
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
K tuples:
   POWER(z0, Cons(z1, z2)) -> c(MULT(z0, power(z0, z2)), POWER(z0, z2))
   MULT(z0, Cons(x1, Nil)) -> c2
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c4_1, c_1, c2_2, c2, c_2


----------------------------------------

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   MULT(z0, Cons(x1, Nil)) -> c2
   POWER(z0, Cons(x1, Nil)) -> c(MULT(z0, Cons(Nil, Nil)))

----------------------------------------

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))), POWER(z0, Cons(x2, Nil)))
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, MULT_2, POWER_2

Compound Symbols:   c4_1, c2_2, c_2


----------------------------------------

(57) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))))
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2)))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, MULT_2, POWER_2

Compound Symbols:   c4_1, c2_2, c_2, c_1


----------------------------------------

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MULT(z0, Cons(x1, Cons(z1, z2))) -> c2(ADD0(z0, add0(z0, mult(z0, z2))), MULT(z0, Cons(z1, z2))) by 
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)), MULT(z0, Cons(x2, Nil)))

----------------------------------------

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)), MULT(z0, Cons(x2, Nil)))
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)), MULT(z0, Cons(x2, Nil)))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c4_1, c_2, c_1, c2_2


----------------------------------------

(61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c4_1, c_2, c_1, c2_2, c2_1


----------------------------------------

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
We considered the (Usable) Rules:none
And the Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(ADD0(x_1, x_2)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(MULT(x_1, x_2)) = [1]
   POL(Nil) = [1]
   POL(POWER(x_1, x_2)) = [1] + x_2
   POL(add0(x_1, x_2)) = x_2
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(mult(x_1, x_2)) = [1] + x_2
   POL(power(x_1, x_2)) = [1] + x_1 + x_2

----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
K tuples:
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c4_1, c_2, c_1, c2_2, c2_1


----------------------------------------

(65) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace POWER(z0, Cons(x1, Cons(x2, Nil))) -> c(MULT(z0, mult(z0, Cons(Nil, Nil)))) by POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil))))
----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
   POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil))))
S tuples:
   ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
K tuples:
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c4_1, c_2, c2_2, c2_1, c_1


----------------------------------------

(67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD0(z0, Cons(z1, z2)) -> c4(ADD0(z0, z2)) by 
   ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2)))

----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
   POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil))))
   ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2)))
S tuples:
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2)))
K tuples:
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c_2, c2_2, c2_1, c_1, c4_1


----------------------------------------

(69) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil))) by MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0))
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil))))
   ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2)))
   MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0))
S tuples:
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2)))
K tuples:
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c_2, c2_2, c_1, c4_1, c2_1


----------------------------------------

(71) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, mult(z0, Nil)))) by POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, Nil)))
----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2)))
   MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0))
   POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, Nil)))
S tuples:
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2)))
K tuples:
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c_2, c2_2, c4_1, c2_1, c_1


----------------------------------------

(73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD0(z0, Cons(z1, Cons(y1, y2))) -> c4(ADD0(z0, Cons(y1, y2))) by 
   ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3))))

----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0))
   POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, Nil)))
   ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3))))
S tuples:
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3))))
K tuples:
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c_2, c2_2, c2_1, c_1, c4_1


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(75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MULT(z0, Cons(z1, Cons(z2, Nil))) -> c2(ADD0(z0, z0)) by 
   MULT(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Cons(z2, Nil))) -> c2(ADD0(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(y1, Cons(y2, Cons(y3, y4)))))

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(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   power(z0, Cons(z1, z2)) -> mult(z0, power(z0, z2))
   power(z0, Nil) -> Cons(Nil, Nil)
   mult(z0, Cons(z1, z2)) -> add0(z0, mult(z0, z2))
   mult(z0, Nil) -> Nil
   add0(z0, Cons(z1, z2)) -> Cons(Cons(Nil, Nil), add0(z0, z2))
   add0(z0, Nil) -> z0
Tuples:
   POWER(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c(MULT(z0, mult(z0, mult(z0, power(z0, z2)))), POWER(z0, Cons(x2, Cons(z1, z2))))
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   POWER(z0, Cons(z1, Cons(z2, Nil))) -> c(MULT(z0, add0(z0, Nil)))
   ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3))))
   MULT(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(z1, Cons(z2, Nil))) -> c2(ADD0(Cons(y1, Cons(y2, Cons(y3, y4))), Cons(y1, Cons(y2, Cons(y3, y4)))))
S tuples:
   MULT(z0, Cons(x1, Cons(x2, Cons(z1, z2)))) -> c2(ADD0(z0, add0(z0, add0(z0, mult(z0, z2)))), MULT(z0, Cons(x2, Cons(z1, z2))))
   ADD0(z0, Cons(z1, Cons(z2, Cons(y2, y3)))) -> c4(ADD0(z0, Cons(z2, Cons(y2, y3))))
K tuples:
   MULT(z0, Cons(x1, Cons(x2, Nil))) -> c2(ADD0(z0, add0(z0, Nil)))
Defined Rule Symbols:   power_2, mult_2, add0_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c_2, c2_2, c_1, c4_1, c2_1